For two functions $$f ( x ) = \frac { 1 } { 3 } x ( 4 - x ) , \quad g ( x ) = | x - 1 | - 1$$ let $S$ denote the area enclosed by their graphs. Find the value of $4 S$. [4 points]

The area enclosed by the curves $y = \sin x + \cos x$ and $y = | \cos x - \sin x |$ over the interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(A) $4 ( \sqrt { 2 } - 1 )$
(B) $2 \sqrt { 2 } ( \sqrt { 2 } - 1 )$
(C) $2 ( \sqrt { 2 } + 1 )$
(D) $2 \sqrt { 2 } ( \sqrt { 2 } + 1 )$

Consider the functions $f , g : \mathbb { R } \rightarrow \mathbb { R }$ defined by
$$f ( x ) = x ^ { 2 } + \frac { 5 } { 12 } \quad \text { and } \quad g ( x ) = \begin{cases} 2 \left( 1 - \frac { 4 | x | } { 3 } \right) , & | x | \leq \frac { 3 } { 4 } \\ 0 , & | x | > \frac { 3 } { 4 } \end{cases}$$
If $\alpha$ is the area of the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : | x | \leq \frac { 3 } { 4 } , 0 \leq y \leq \min \{ f ( x ) , g ( x ) \} \right\}$$
then the value of $9 \alpha$ is $\_\_\_\_$ .

The area (in sq. units) of the region $A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\, 0 \leq x \leq 2\}$, where $[t]$ denotes the greatest integer function, is:
(1) $\frac{8}{3}\sqrt{2} - \frac{1}{2}$
(2) $\frac{4}{3}\sqrt{2} + 1$
(3) $\frac{8}{3}\sqrt{2} - 1$
(4) $\frac{4}{3}\sqrt{2} - \frac{1}{2}$

The area of the region $\{ ( x , y ) : x ^ { 2 } \leq y \leq | x ^ { 2 } - 4 | , y \geq 1 \}$ is
(1) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } - 1 )$
(2) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } - 1 )$

The area of the region $\{(x, y): x^2 \leq y \leq |x^2 - 4|, y \geq 1\}$ is
(1) $\frac{4(\sqrt{5}-1)}{3} + 4$
(2) $\frac{4(\sqrt{5}-1)}{3} + 2$
(3) $\frac{2(\sqrt{5}-1)}{3} + 4$
(4) $\frac{2(\sqrt{5}-1)}{3} + 2$

The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is :
(1) $\frac { 4 } { 3 }$
(2) 1
(3) $\frac { 2 } { 3 }$
(4) $\frac { 8 } { 3 }$

The area of the region $\left\{(x, y) : x^2 + 4x + 2 \leq y \leq |x+2|\right\}$ is equal to
(1) 7
(2) 5
(3) $24/5$
(4) $20/3$

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq 2 | x | + 1,0 \leq y \leq x ^ { 2 } + 1 , | x | \leq 3 \right\}$ is
(1) $\frac { 80 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) $\frac { 17 } { 3 }$